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A132411
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a(0) = 0, a(1) = 1 and a(n) = n^2 - 1 with n>=2.
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11
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0, 1, 3, 8, 15, 24, 35, 48, 63, 80, 99, 120, 143, 168, 195, 224, 255, 288, 323, 360, 399, 440, 483, 528, 575, 624, 675, 728, 783, 840, 899, 960, 1023, 1088, 1155, 1224, 1295, 1368, 1443, 1520, 1599, 1680, 1763
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Sequence allows us to find X values of the equation: X^3 - (X + 1)^2 + X + 2 = Y^2.
To prove that X = 1 or X = n^2 - 1: Y^2 = X^3 - (X + 1)^2 + X + 2 = X^3 - X^2 - X + 1 = (X + 1)(X^2 - 2X + 1) = (X + 1)*(X - 1)^2 it means: X = 1 or (X + 1) must be a perfect square, so X = 1 or X = n^2 - 1 with n>=1. which gives: (X, Y) = (0, 1) or (X, Y) = (1, 0) or (X, Y) = (n^2 - 1, n*(n^2 - 2))with n>=2.
An equivalent technique of integer factorization would work for example for the equation X^3+3*X^2-9*X+5=(X+5)(X-1)^2=Y^2, looking for perfect squares of the form X+5=n^2. Another example is X^3+X^2-5*X+3=(X+3)*(X-1)^2=Y^2 with solutions generated from perfect squares of the form X+3=n^2. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 20 2007
a(n) = A170949(A002522(n-1)) for n>0. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 08 2010]
Sum of possible divisors of a prime number up to its square root. [From Odimar Fabeny (aifab(AT)yahoo.com.br), Aug 25 2010]
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FORMULA
| a(n) = A005563(n-1), n>1.
G.f.: x+x^2*(-3+x)/(-1+x)^3 . - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 20 2007
Starting (1, 3, 8, 15, 24,...) = binomial transfor of [1, 2, 3, -1, 1, -1,...] - Gary W. Adamson (qntmpkt(AT)yahoo.com), May 12 2008
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EXAMPLE
| 0^3 - 1^2 + 2 = 1^2, 1^3 - 2^2 + 3 = 0^2, 3^3 - 4^2 + 5 = 4^2.
For P(n) = 29 we have sqrt(29) = 5.3851 then possible divisors are 3 and 5; for P(n) = 53 we have sqrt(53) = 7.2801 then possible divisors are 3, 5 and 7. [From Odimar Fabeny (aifab(AT)yahoo.com.br), Aug 25 2010]
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CROSSREFS
| Cf. A028560, A005563.
Sequence in context: A086959 A083656 A013648 * A131386 A005563 A067998
Adjacent sequences: A132408 A132409 A132410 * A132412 A132413 A132414
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KEYWORD
| nonn,easy
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AUTHOR
| Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Nov 12 2007
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EXTENSIONS
| Simplified definition. - N. J. A. Sloane, Sep 05 2010
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